 reserve n,k for Nat;
 reserve L for comRing;
 reserve R for domRing;
 reserve x0 for positive Real;

theorem Th20:
  for k be Nat,f be Element of the carrier of Polynom-Ring INT.Ring
    st ((Der1(INT.Ring))|^1).(f|^1) = 1.Polynom-Ring INT.Ring holds
     ((Der1(INT.Ring))|^k).(f|^k) = (k!)*(1.Polynom-Ring INT.Ring)
     proof
       set L = Polynom-Ring INT.Ring;
       set D = Der1(INT.Ring);
       let k be Nat, f be Element of the carrier of L;
       assume
A1:    (D|^1).(f|^1) = 1.L;
A2:    f|^0 = 1_L by BINOM:8 .= 1.L;
A3:    (eta(k,k)) = (k!)/(0!) by XREAL_1:232 .= k! by NEWTON:12;
       per cases;
         suppose
           k <> 0; then
           1 <= k <= k by NAT_1:14; then
           (D|^k).(f|^k) = (k!)*(f|^(k -'k)) by A3,Th19,A1
           .= (k!)*(1.L) by A2,XREAL_1:232;
           hence thesis;
         end;
         suppose
A5:        k = 0;
           D|^0 = id L by VECTSP11:18;
           hence thesis by A2,A5,NEWTON:12,BINOM:13;
         end;
       end;
